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A Criterion for a Polynomial to Factor Completely over the Integers

A Criterion for a Polynomial to Factor Completely over the Integers A CRITERION FOR A POLYNOMIAL TO FACTOR COMPLETELY OVER THE INTEGERS WALTER FEITf AND ELMER REES Let p be a prime, let F be the field of p elements and F be the algebraic closure of F . If q = p\ let F be the unique subfield of F of cardinality q. p q If / =/(*) is a monic polynomial with coefficients in F and if A: is a positive integer, let s (f) denote the sum of the A--th powers of all the roots of/(x) (counted with multiplicities). By Newton's formulas s (f) is a polynomial with coefficients in F of the coefficients of f(x). THEOREM 1. Let f(x) be a monic polynomial with coefficients in F. Suppose that f(x) = g(x)h(x ) for monic polynomials g(x), h(x) such that the roots of g(x) all lie in Fq for some q. Then s (f) = s _ (f) for all positive integers k. k k+q l Conversely suppose that f(x) has exactly n distinct nonzero roots a ...,a in F it n which occur with a multiplicity not divisible by p. If for some q = p' **(/ ) = ** -i(/ ) M l^k^n (*) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

A Criterion for a Polynomial to Factor Completely over the Integers

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/10.2.191
Publisher site
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Abstract

A CRITERION FOR A POLYNOMIAL TO FACTOR COMPLETELY OVER THE INTEGERS WALTER FEITf AND ELMER REES Let p be a prime, let F be the field of p elements and F be the algebraic closure of F . If q = p\ let F be the unique subfield of F of cardinality q. p q If / =/(*) is a monic polynomial with coefficients in F and if A: is a positive integer, let s (f) denote the sum of the A--th powers of all the roots of/(x) (counted with multiplicities). By Newton's formulas s (f) is a polynomial with coefficients in F of the coefficients of f(x). THEOREM 1. Let f(x) be a monic polynomial with coefficients in F. Suppose that f(x) = g(x)h(x ) for monic polynomials g(x), h(x) such that the roots of g(x) all lie in Fq for some q. Then s (f) = s _ (f) for all positive integers k. k k+q l Conversely suppose that f(x) has exactly n distinct nonzero roots a ...,a in F it n which occur with a multiplicity not divisible by p. If for some q = p' **(/ ) = ** -i(/ ) M l^k^n (*)

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jul 1, 1978

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